are equal. Any interpolator would give a constant 
geoid value even under the mountain. Obviously, 
this is wrong since the geoidal signal due to the 
mountain is completely missed. On the contrary, by 
introducing the N rt c component, this signal is 
correctly modelled and taken into account. To 
quantify the mismodelling that can be introduced by 
an improper handling of the topography, a very 
simple example can be shown. 
Two parallelepipeda are combined to simulate the 
geoidal effect of a mountain and the undulation 
values are supposed to be sampled in points A and 
B (see Figure 2). 
Figure 2. The simulated mountain 
By assuming a density of 2.67 g/cm**3 and using 
the closed formula giving the potential of a 
parallelepiped (W.D. Mac Millan, 1958), a common 
value of 10 cm can be computed for the terrain 
component N rtc . Hence, any interpolator would give 
the same value in C, placed on top of the mountain, 
while the correct directly computed value is 16 cm. 
So, a bias of 6 cm is introduced by disregarding the 
topographic signal, even considering a quite small 
mountain. 
Thus, this simple example proves that with an 
unhomogeneous distribution of data, biased 
estimations can occur indipendently from the used 
interpolator. 
In the following, numerical tests will be carried out to 
define the optimal procedure and the best 
interpolation method for geoid prediction from 
GPS/levelling derived undulations. 
2. NUMERICAL TESTS ON GEOID 
INTERPOLATION 
2.1 The geoid simulation procedure 
Simulated geoid data in selected areas have been 
computed according to the following scheme: 
1) geoid values on a regular 3’ grid; 
2) scattered geoid values (point positions as 
derived from real data distributions); 
3) geoid values on a regular 5’ grid. 
The low frequency component has been derived 
from the OSU91A geopotential model (R. Rapp, 
1997), the RTC effect has been computed by using 
the Italian DTM data base (M.T. Carrozzo et 
al,1982) and the residual geoid signal N r has been 
obtained from the ITALGE095 solution (R. Barzaghi 
et al.,1996). For each selected area, two different 
interpolation methods have been applied to predict 
from data sets 1) and 2) on 3). So, data set 3) is the 
target data set while 1) and 2) are the input data 
bases. Using data base 1) to predict on 3) is quite 
unrealistic, since, as it is pointed out before, real 
data are not usually regularly distributed. 
Nevertheless, this computation allows testing of 
different interpolation methods in an optimal input 
data case. On the contrary, prediction from 2) on 3) 
presents a more realistic situation, where data 
distribution is critical in determining the estimated 
values. 
For both cases, predictions have been computed 
using: 
a) the total geoid signal 
N T =N M +N rtc +N r 
b) the geoid data after low frequency component 
reduction 
N r =N t -N m 
c) the residual geoid obtained by removing both 
low and high frequency geoid signals 
N r =N T -N M -N rtc . 
This is done to account not only for interpolation 
method and input data distribution on the final result 
but also for spectral properties of input geoidal data. 
2.2 Test area (A) 
The area centered on the Trento - Bolzano 
provinces has been selected as the first test area. 
Data were considered in a window with boundaries 
45.5 <(p <47 10.2 < A < 11.7 
This is a rough topography area in the eastern Alps; 
in Figure 3 the total geoid signal is plotted and the